Two Functional Inequalities

by Mathdreams, Apr 6, 2025, 1:34 PM

Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x) \le x^3$ and $f(x + y) \le f(x) + f(y) + 3xy(x + y)$ for any real numbers $x$ and $y$ .

(Miroslav Marinov, Bulgaria)

functional inequalities

algebra

Stereotypical Diophantine Equation

by Mathdreams, Apr 6, 2025, 1:32 PM

Find all solutions in the nonnegative integers to $2^a3^b5^c7^d - 1 = 11^e$ .

(Shining Sun, USA)

number theory

Diophantine equation

Two Orthocenters and an Invariant Point

by Mathdreams, Apr 6, 2025, 1:30 PM

Let $\triangle{ABC}$ be a triangle, and let $P$ be an arbitrary point on line $AO$ , where $O$ is the circumcenter of $\triangle{ABC}$ . Define $H_1$ and $H_2$ as the orthocenters of triangles $\triangle{APB}$ and $\triangle{APC}$ . Prove that $H_1H_2$ passes through a fixed point which is independent of the choice of $P$ .

(Kritesh Dhakal, Nepal)

geometry

Geometry

by youochange, Apr 6, 2025, 11:27 AM

m:}
Let $\triangle ABC$ be a triangle inscribed in a circle, where the tangents to the circle at points $B$ and $C$ intersect at the point $P$ . Let $M$ be a point on the arc $AC$ (not containing $B$ ) such that $M \neq A$ and $M \neq C$ . Let the lines $BC$ and $AM$ intersect at point $K$ . Let $P'$ be the reflection of $P$ with respect to the line $AM$ . The lines $AP'$ and $PM$ intersect at point $Q$ , and $PM$ intersects the circumcircle of $\triangle ABC$ again at point $N$ .

Prove that the point $Q$ lies on the circumcircle of $\triangle ANK$ .

This post has been edited 1 time. Last edited by youochange, 4 hours ago
Reason: Y

geometry

Beautiful problem

by luutrongphuc, Apr 4, 2025, 5:35 AM

Let triangle $ABC$ be circumscribed about circle $(I)$ , and let $H$ be the orthocenter of $\triangle ABC$ . The circle $(I)$ touches line $BC$ at $D$ . The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$ . Let $J$ be the midpoint of $HI$ , and let the line $DJ$ meet $(I)$ again at $X$ . The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$ . Prove that $ST$ is tangent to $(I)$ .

geometry

Common tangent to diameter circles

by Stuttgarden, Mar 31, 2025, 1:06 PM

The cyclic quadrilateral $ABCD$ , inscribed in the circle $\Gamma$ , satisfies $AB=BC$ and $CD=DA$ , and $E$ is the intersection point of the diagonals $AC$ and $BD$ . The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$ . Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$ .

geometry

Spain

Nordic 2025 P1

by anirbanbz, Mar 25, 2025, 12:32 PM

Let $n$ be a positive integer greater than $2$ . Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ satisfying:
$(f(x+y))^{n} = f(x^{n})+f(y^{n}),$ for all integers $x,y$

Problem 1 of the HMO 2025

by GreekIdiot, Feb 22, 2025, 12:32 PM

Let $P(x)=x^4+5x^3+mx^2+5nx+4$ have $2$ distinct real roots, which sum up to $-5$ . If $m,n \in \mathbb {Z_+}$ , find the values of $m,n$ and their corresponding roots.

This post has been edited 1 time. Last edited by GreekIdiot, Feb 22, 2025, 3:33 PM

ISI 2019 : Problem #7

by integrated_JRC, May 5, 2019, 6:00 PM

Let $f$ be a polynomial with integer coefficients. Define $a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$ and $~a_n = f(a_{n-1})$ for $n \geqslant 3$ .

If there exists a natural number $k \geqslant 3$ such that $a_k = 0$ , then prove that either $a_1=0$ or $a_2=0$ .

isi

Indian Statistical Institute

Some really bad rings

by math_explorer, Sep 26, 2017, 12:36 AM

Consider $\mathbb{Q}[x_1, x_2, x_3, \ldots]$ where you adjoin infinitely many free variables. This has infinite Krull dimension because ideals $(x_1, x_2, \ldots, x_n)$ are all prime. It's also local; the ideal generated by all the variables is the unique maximal ideal.

Consider $\mathbb{Q}[x, x^{\omega-n}\text{ for all }n \in \mathbb{Z}]$ ; think of $\omega$ like an infinity. It is local because $(x)$ is the unique maximal ideal. The ideal generated by all $x^{\omega-n}$ is prime; that maximal ideal $(x)$ divides it infinitely many times.

Consider $\mathbb{Z} + x\mathbb{Q}[x]$ , the ring of polynomials with integer constant coefficient. Each integer prime or prime integer or something $p$ generates a maximal ideal $(p)$ . The intersection of those infinitely many maximal ideals is the prime ideal $(x)$ .

algebra

abstract algebra

algebra is hard

by math_explorer, Jul 6, 2017, 4:23 AM

http://mit.edu/bpchen/Public/2017/reconstruction.png

algebra

Find an angle

by socrates, Nov 5, 2016, 10:48 PM

Let $ABCD$ be a parallelogram such that $\angle BAD = 60^{\circ}.$ Let $K$ and $L$ be the midpoints of $BC$ and $CD,$ respectively. Assuming that $ABKL$ is a cyclic quadrilateral, find $\angle ABD.$

geometry

side bash

Submit

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